Math 200 University of Connecticut

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1 RELATIVISTIC ADDITION AND REAL ADDITION KEITH CONRAD Math 200 University of Connecticut Date: Aug. 31, 2005.

2 RELATIVISTIC ADDITION AND REAL ADDITION 1 1. Introduction For three particles P, Q, R travelling on a straight line, let v P Q be the (relative) velocity of P as measured by Q, and define v QR, v P R similarly. According to classical mechanics, the velocity v of a particle moving on a line can be any real number, and relative velocities add by the simple formula (1.1) v P R = v P Q + v QR. On the other hand, the special theory of relativity says velocities are restricted to a bounded range, c < v < c, where c is the speed of light (whose value of course depends on the choice of units, and it is convenient to choose them so c = 1, but we won t do that.) The relativistic addition formula for velocities is: v P Q + v QR (1.2) v P R = 1 + (v P Q v QR /c 2 ). Example 1.1. If v P Q = 3c and v 4 QR = 1 c then a comparison of the right sides of (1.1) and 2 (1.2) gives v P Q + v QR = 5 4 c > c, v P Q + v QR 1 + (v P Q v QR /c 2 ) = c < c. While the first formula gives a value which exceeds c, the second formula gives a value which is less than c. Although (1.1) and (1.2) appear different, when both velocities are small they are approximately the same. Indeed, if v P Q, v QR c then v P Q v QR /c 2 0 and the denominator in (1.2) is nearly 1, so (1.2) is approximately equal to its numerator, which is the right side of (1.1). If only one velocity is small compared to c, then the difference between (1.1) and (1.2) can still be noticed if we think about the relative error compared to the size of the smaller velocity. Table 1 gives in the last column the difference between classical and relativistic formulas for v P R. Note there is significant relative error not only in the first row, when both v P Q and v QR are substantial fractions of the speed of light, but even in the second and third rows, when only v P Q is a substantial fraction of c. Specifically, for small v QR /c the fourth column entry is about.5625v QR = (9/16)v QR, which is a not insubstantial fraction of v QR. v P Q v QR (1.2) (1.1) (1.2) (3/4)c (1/2)c (10/11)c.341c (3/4)c (1/100)c.75434c.00566c (3/4)c (1/1000)c c c Table 1. Classical and Relativistic Comparisons We can use a Taylor expansion to understand why (1.2) noticeably deviates from (1.1) as soon as just one of the velocities is not small compared to c. For v, w ( c, c), take v to be fixed and think of h(w) = v w = (v + w)/(1 + vw/c 2 ) as a function of w. For w/c small, a Taylor expansion at 0 yields ) v w h(0) + h (0)w = v + (1 v2 w. c 2

3 2 KEITH CONRAD If v/c is also small, the coefficient of w is about 1, so v w v + w. But if v/c is not small, e.g., v = 3 c, then we see a deviation from the classical velocity addition rule v + w by an 4 error of around (v/c) 2 w. This explains the error 9 v 16 QR in the table above, where v = 3c. 4 Our goal in this paper is not to focus on the differences between (1.1) and (1.2), but on their similarities through common algebraic properties. The classical model for velocity allows all real numbers as velocities and combines them under addition. Special relativity involves velocities in an interval ( c, c) and combines them by the formula (1.3) v w := v + w 1 + vw/c 2. The following algebraic properties of the operation are similar to properties of ordinary addition on R: Closure, i.e., v, w ( c, c) v w ( c, c). Identity for : 0 v = v 0 = v for v ( c, c). Inverse of any v under is v: v v = v v = 0. Associativity: (v w) x = v (w x) for v, w, x ( c, c). It is left to the reader to check these, of which the first and fourth are the only ones with much content. Note usual addition is not closed on ( c, c), as we saw in Example 1.1. In addition to the operation on ( c, c), the operation of multiplication on the interval (0, ) of positive real numbers also has the four properties above (where the identity is 1 and the inverse of x is 1/x), and moreover we know the logarithm turns multiplication on (0, ) into addition on R: log(xy) = log x + log y, where log means the natural logarithm (or use whatever logarithmic base you want). It turns out that there is an analogous relativistic logarithm L(v) defined on numbers v ( c, c) which turns on ( c, c) into addition on R: L(v w) = L(v) + L(w). In other words, relativistic addition on ( c, c) is just a disguised form of ordinary addition on R, and the relativistic logarithm (which we have not written down yet as an explicit formula) unmasks this relationship. In the remainder of this paper we will identify the relativistic logarithm in Section 2 and then show in Section 3, quite generally, that any reasonable addition-like operation on any interval of real numbers has its own special log-like function which reveals the operation to be ordinary addition in a hidden form. 2. Rescaling with the hyperbolic tangent The formula for v w in (1.3) is reminiscent of the addition formula for the tangent function: tan x + tan y tan(x + y) = 1 (tan x)(tan y). However, there is a minus sign in the denominator here where there is a plus sign in the denominator of (1.3). The hyperbolic tangent is better. It is defined by the formula tanh x = ex e x e x + e x,

4 RELATIVISTIC ADDITION AND REAL ADDITION 3 where x is any real number. (A discussion of the hyperbolic functions is in Appendix A. In particular, their strong similarities to the classical trigonometric functions are seen there using calculus.) The values of tanh x lie in the interval ( 1, 1) and a little algebra shows tanh(x + y) = tanh x + tanh y 1 + (tanh x)(tanh y). This is exactly like (1.3), up to factors of c. Let s scale the hyperbolic tangent by c to make its values lie in ( c, c). The function ϕ(x) := c tanh(x) has values in ( c, c) and (2.1) ϕ(x + y) = ϕ(x) ϕ(y). This should be considered an analogue for on ( c, c) of the exponential rule e x+y = e x e y. What we actually care more about is not a way of transforming + into but into + (turn the more complicated operation into an easier operation). For this, we should use the inverse of the hyperbolic tangent. To find it, we carry out some algebra: if y = tanh x (so 1 < y < 1), we want to write x in terms of y. Since clearing denominators gives Collecting the x-terms on the left, Dividing and taking care of sign issues, y = ex e x e x + e x = e2x 1 e 2x + 1, ye 2x + y = e 2x 1. e 2x (y 1) = 1 y. e 2x = 1 + y 1 y. Taking natural logarithms of both sides and dividing by 2, (2.2) x = 1 ( ) 1 + y 2 log. 1 y Since y = tanh x, (2.2) gives us the inverse hyperbolic tangent function: tanh 1 (y) = 1 ( ) 1 + y 2 log 1 y when 1 < y < 1. The c-scaled version of the inverse hyperbolic tangent, which is defined on ( c, c) instead of on ( 1, 1), is L(v) = 1 ( ) 1 + v/c 2 log. 1 v/c Since L(ϕ(x)) = x, where ϕ(x) = c tanh x, applying L to both sides of (2.1) shows L turns into ordinary addition, just as the usual logarithm turns multiplication of positive numbers into ordinary addition: L(v w) = L(v) + L(w).

5 4 KEITH CONRAD Thus by the clever transformation L(v), which is essentially the inverse of the hyperbolic tangent of v, we can replace velocities v ( c, c) by rescaled velocities L(v) R and this converts on ( c, c) into ordinary addition of real numbers. Example 2.1. Using the values from Example 1.1, if v = 3c and w = 1c then L(v) = 1 log and L(w) = 1 10 log 3, while L(v w) = L( c) = 1 log 21 = L(v) + L(w) More general addition laws Rather than just looking at on ( c, c), let s consider any open interval I R with an addition law. That is, has the following four properties: Closure. x, y I = x y I. Identity. There is u I such that for any x I, x u = u x = x. Inverses. For any x I there is some i(x) I such that x i(x) = i(x) x = u. Associativity. For x, y, z I, (x y) z = x (y z). Example 3.1. Take I = (0, ) and x y = xy, so addition is really ordinary multiplication. It has the four properties above. It will be useful to write the operation x y in the notation of a function of two variables: F (x, y) = x y. For example, the classical and relativistic velocity addition formulas are F (v, w) = v + w, I = R; F (v, w) = v + w, I = ( c, c). 1 + vw/c2 The above properties of take the following form in terms of F : x, y I F (x, y) I. There is u I such that for any x I, F (x, u) = F (u, x) = x. For any x I there is some i(x) I such that F (x, i(x)) = F (i(x), x) = u. For x, y, z I, F (F (x, y), z) = F (x, F (y, z)). Our main goal is to prove the following theorem, which says that under a suitable differentiability assumption this addition law on I admits a logarithm that turns it into ordinary addition on R. Theorem 3.2. If F (x, y) = x y has continuous partial derivatives in x and y then there is a differentiable function f : I R that converts on I to ordinary addition on R. That is, f is a differentiable bijection and f(x y) = f(x) + f(y). Theorem 3.2 will be proved by giving an explicit recipe for f. But we don t just want to write down a formula and blindly check it works. We want to understand where it comes from. Our treatment follows [3, pp ]. To discover f, rewrite the desired equation f(x y) = f(x) + f(y) as f(f (x, y)) = f(x) + f(y). Let s differentiate this equation with respect to x: f (F (x, y))f 1 (x, y) = f (x), where we write F 1 (x, y) for the derivative with respect to the first variable x. Similarly we will write F 2 (x, y) for the derivative with respect to the second variable y. Setting x = u (the identity for ), F (x, y) = F (u, y) = u y = y, so f (y)f 1 (u, y) = f (u).

6 Let s now solve for f (y) and integrate: (3.1) f(y) = RELATIVISTIC ADDITION AND REAL ADDITION 5 y u f (u) F 1 (u, t). This is a formula for the rescaling function f. The constant f (u) is just an arbitrary factor, which we will choose to be 1 when we finally make a definition of f(x). When we integrated in (3.1), we didn t introduce an additive constant since we want f(u) = 0 (just like log 1 = 0) and the integral formula in (3.1) has this feature already. Of more pressing interest is the validity of dividing by F 1 (u, t) in (3.1). Why is it never zero? Lemma 3.3. For any t I, F 1 (u, t) > 0. Proof. We differentiate the associative law, x (y z) = (x y) z, with respect to x. Using the F -notation instead of the -notation, recall associativity of says F (F (x, y), z) = F (x, F (y, z)). Differentiating both sides of this with respect to x, Setting x = u, F (x, y) becomes y: F 1 (F (x, y), z)f 1 (x, y) = F 1 (x, F (y, z)). (3.2) F 1 (y, z)f 1 (u, y) = F 1 (u, F (y, z)) = F 1 (u, y z). So if F 1 (u, y) = 0 for some y, then F 1 (u, y z) = 0 for any z. F 1 (u, u) = 0. But this is not true: (3.3) F (x, u) = x for all x F 1 (x, u) = 1 for all x F 1 (u, u) = 1. Choose z = i(y) to get So F 1 (u, y) is nonzero for every y. Since it equals 1 at y = u, it must always be positive by the Intermediate Value Theorem (F 1 is continuous). This lemma allows us to divide by F 1 (u, t) for any t I, and we will do this often without explicitly appealing to the lemma each time. Since 1/F 1 (u, t) is continuous in t, hence integrable, we are justified in making the following definition, for any x in the interval I: (3.4) f(x) def = u F 1 (u, t). By the fundamental theorem of calculus, f is differentiable and (3.5) f (x) = 1 F 1 (u, x). In particular, f (u) = 1 by (3.3). With the choice of (3.4) for our rescaling function we now prove our main result Theorem 3.2. Proof. We need to check two things: f(f (x, y)) = f(x) + f(y). f : I R is a bijection. For the first item, fix y I. We consider the x-derivatives of the two functions f(f (x, y)), f(x) + f(y).

7 6 KEITH CONRAD By (3.5), the derivative of the first function is f (F (x, y))f 1 (x, y) = F 1 (x, y) F 1 (u, F (x, y)). Does this equal the x-derivative of the second function, namely f (x) = 1/F 1 (u, x)? Setting them equal, we want to consider: F 1 (x, y)f 1 (u, x)? = F 1 (u, F (x, y)). This is just (3.2) with x, y, z relabelled as u, x, y. Therefore f(f (x, y)) and f(x) + f(y) have equal x-derivatives for all x. Since they are equal at x = u, they are equal for all x. This verifies the first item. For the second item, bijectivity of f : I R, since f (y) = 1/F 1 (u, y) > 0 we see f is increasing, hence injective. To show surjectivity, note f(i) is an interval by continuity. Choose x I, x u. Since f(x) + f(i(x)) = f(x i(x)) = f(u) = 0, f(x) and f(i(x)) have opposite sign, one positive and the other negative. For any positive integer n, n times n times {}}{{}}{ f( x x) = nf(x), f( i(x) i(x)) = nf(i(x)), As n, one of these tends to and the other to. Since f(i) is an interval, we must have f(i) = R. Corollary 3.4. When the operation x y = F (x, y) has continuous partial derivatives, it is commutative. In particular, F 1 (x, y) = F 2 (y, x) for all x, y I. Proof. Commutativity was never used in the proof of Theorem 3.2, so commutativity of ordinary addition on R implies commutativity of on I. That is, in terms of the rescaling function ( logarithm ) f from I to R, for any x and y in I we have f(x y) = f(x) + f(y) = f(y) + f(x) = f(y x), so x y = y x since f is injective. Now differentiate both sides of the formula x y = y x with respect to x, rewriting this commutativity formula first as F (x, y) = F (y, x) to make the notation easier to use in the context of calculus. We obtain F 1 (x, y) = F 2 (y, x). Corollary 3.4 says there is no way to define a non-commutative addition law on an interval of R, granting reasonable differentiability assumptions on x y as a function of x and y. (The same conclusion holds if we grant only assumptions of continuity for x y as a function of x and y, but the proof then becomes more difficult. See [2].) For comparison, once we move beyond intervals and allow two degrees of freedom, non-commutative addition laws are possible. For instance, multiplication of the matrices ( a 0 1 b ) where a > 0 and b R is non-commutative but still shares the four basic properties from the start of this section. Corollary 3.5. The rescaling function f determines the operation by x y = f 1 (f(x) + f(y)). Proof. Apply f 1 to both sides of f(x y) = f(x) + f(y).

8 RELATIVISTIC ADDITION AND REAL ADDITION 7 Since f is determined by the function F 1 (u, t) (Theorem 3.2), and the operation x y is determined by f (Corollary 3.5), we see the operation is encoded in the function F 1 (u, t) = F 2 (u, t). Choosing coordinates so u = 0, the function F 1 (u, x) = F 1 (0, x) appears in the first term of the Taylor expansion at y = u of x y = F (x, y) for small y: (3.6) x y = F (x, y) F (x, 0) + F 2 (x, 0)y = x + F 1 (0, x)y. Since F 1 (0, 0) = 1, for small x and y we get from (3.6) that x y x + y. But for small y and not-so-small x there is a deviation of x y from the simple law x + y, measured by the function F 1 (0, x). This deviation of x y from x + y for any x and small y, which we already noticed in the example of relativistic addition in Table 1, can be used through its appearance in the function f(x) to reconstruct the operation x y for any x and any y. Let s look at some examples, to see which functions rescale various addition laws on an interval to ordinary addition on the real line. Example 3.6. If F (x, y) = x + y on I = R, then u = 0, F 1 (0, x) = 1, and f(x) = Example 3.7. If F (v, w) = v w = and 0 F 1 (0, t) = = x. 0 v + w 1 + vw/c on ( c, c), then u = 0, F 1(0, v) = 1 v2, 2 c 2 v v f(v) = 0 F 1 (0, t) = c2 0 c 2 t = c v ( c t + 1 ) = c ( ) 1 + v/c c + t 2 log. 1 v/c This is the same as the rescaling function L(v) we met at the beginning, up to a factor of c. Of course if the rescaling function f(x) in Theorem 3.2 is multiplied by a nonzero constant, it has the same relevant properties (except f (u) 1.) Example 3.8. As a third and final example, consider F (x, y) = xy on I = (0, ). This is the set of positive real numbers under multiplication. (The set of all nonzero real numbers is not an interval, and the reader should check to see where the proof of Theorem 3.2 breaks down in this case.) Here u = 1 and F 1 (1, x) = x, so Theorem 3.2 tells us that a rescaling function which converts multiplication on (0, ) to addition on R is f(x) = 1 F 1 (1, t) = 1 t = log x. Of course we already knew that log(xy) = log x + log y, but it is still interesting to see how we have rediscovered the logarithm by applying calculus to the general algebraic problem of transforming addition laws on intervals into ordinary addition on the real line. Appendix A. Hyperbolic Functions The hyperbolic trigonometric functions play the same role relative to x 2 y 2 = 1 that the circular trigonometric functions play relative to x 2 + y 2 = 1. We will focus on the hyperbolic analogues of the three basic functions sin t, cos t, and tan t. Definition A.1. For any real number t, the hyperbolic sine and hyperbolic cosine of t are the numbers sinh t = et e t, cosh t = et + e t. 2 2

9 8 KEITH CONRAD The hyperbolic tangent of t is the ratio tanh t = sinh t cosh t = et e t e t + e t. By some algebra, cosh 2 t sinh 2 t = 1. Therefore (cosh t, sinh t) lies on the hyperbola x 2 y 2 = 1. This is why the functions are called hyperbolic. The hyperbola has two branches, one to the right of the y-axis and the other to the left of the y-axis. Since cosh t > 0 for all t, the point (cosh t, sinh t) always lies on the right branch of the hyperbola. Remark A.2. The hyperbolic cosine gives the shape of a freely hanging cable and both sinh t and cosh t are used in formulas for isometries of spacetime [1]. As t runs through the real numbers, these hyperbolic functions doesn t behave like their circular analogues at all: they are not periodic, cosh t and sinh t are not bounded (indeed, their values explode when t ± ), cosh t has no zeros, and tanh t has no asymptotes. What makes them similar is revealed by calculus: their power series expansions are quite close to those of sin t, cos t, and tan t. Recall the series for sin t and cos t: sin t = t t3 3! + t5 5! = ( 1) k t 2k+1 t2, cos t = 1 (2k + 1)! 2! + t4 4! = ( 1) k t2k (2k)!. k 0 k 0 Using the power series for e t, we can compute the series for sinh t and cosh t. Starting with e t = 1 + t + t2 2! + t3 3! + t4 4! + t5 5! +, e t = 1 t + t2 2! t3 3! + t4 4! t5 5! +, we add and subtract and divide in each case by 2: sinh t = t + t3 3! + t5 5! + = k 0 t 2k+1 t2, cosh t = 1 + (2k + 1)! 2! + t4 4! + = k 0 t 2k (2k)!. These are just like the series for sin t and cos t except for the alternating signs. Similarly, tan t = t + t t t t , tanh t = t t t t t References [1] B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, Modern Geometry Methods and Applications, Part I: The Geometry of Surfaces, Transformation Groups, and Fields, Springer-Verlag, New York, [2] I. Kaplansky, Lie Algebras and Locally Compact Groups, Univ. of Chicago Press, Chicago, [3] L. Bowden and M. M. Schiffer, The Role of Mathematics in Science, Mathematical Assoc. of America, Washington, D. C., 1984.